Show that if $(x_1,x_2)$ is defined to be $\{\{x_1\},\{x_1,x_2\}\}$ then $(x_1,x_2)=(y_1,y_2)$ iff $x_1=y_1$ and $x_2=y_2$ My Work:
If you take the cartesian product of any set with two arbitrary elements $a$ and $b$, and the resulting set is $\{\{x_1\},\{x_1,x_2\}\}$, then the only possible values for $a$ and $b$ are $x_1$ and $x_2$ by definition of the cartesian product.
This answer seems overly trivial and I think I'm doing something wrong. Please help!
 A: Lemma:  Denote the cardinality of a set,$A$, by $|A|$.  Suppose $A$ and $B$ are sets with $|A| = 2$ and $|B|=1$, then $A\not = B$. 
Proof:  Let $A=\{a_1,a_2\}$ with $a_1 \not = a_2$, and $B=\{b_1\}$, then if $b_1\not \in A$ we have $A\not = B$ and we are done, so assume that $b_1\in A$.  Without loss of generality we can say $b_1=a_1$.  But in this case we have $a_2\not\in B$ so $A\not = B$.
First of all, it is easy to see that if $x_1 = y_1$ and $x_2 = y_2$ we have
$$
\left\{ \{x_1\}, \{x_1,x_2\}\right\} = \left\{ \{y_1\}, \{y_1,y_2\}\right\},
$$
so we have the backward implication.  For the forward implication we have two cases, for the case $x_1=x_2$ we have
\begin{align*}
  (x_1,x_2) = (y_1,y_2)&  \Leftrightarrow (x_1,x_1) = (y_1,y_2)\\
  &\Leftrightarrow \left\{\{x_1\}, \{x_1,x_1\}  \right\} = \left\{ \{ y_1\}, \{y_1,y_2\} \right\}\\
  &\Leftrightarrow \left\{ \{x_1\}, \{x_1\} \right\} = \left\{\{y_1\}, \{y_1,y_2\} \right\}\\
  &\Leftrightarrow \left\{ \{x_1\} \right\} = \left\{ \{y_1\}, \{y_1,y_2\} \right\}\\
  &\Rightarrow x_1 = y_1\ \mathrm{and\ } \{x_1\} = \{y_1,y_2\}\\
  &\Rightarrow x_1 = y_1\ \mathrm{and\ } x_1=y_1=y_2\\
  &\Rightarrow x_1 = y_1\ \mathrm{and\ } x_2=y_2.
\end{align*}
For the case $x_1\not= x_2$ we have
\begin{align*}
(x_1,x_2) = (y_1,y_2) &\Leftrightarrow \left\{ \{ x_1\}, \{x_1,x_2\} \right\}= \left\{ \{y_1\}, \{y_1,y_2\} \right\}\\ 
&\Leftrightarrow \{x_1\} = \{y_1\}\ \mathrm{and\ } \{x_1,x_2\} = \{y_1,y_2\}\\
&\Leftrightarrow x_1 = y_1\ \mathrm{and\ } \{x_1,x_2\} = \{y_1,y_2\}\\
&\Rightarrow x_1 = y_1\ \mathrm{and\ } \{x_1,x_2\} = \{x_1,y_2\}\\
&\Rightarrow x_1 = y_1\ \mathrm{and\ } x_2 = y_2.
\end{align*}
